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It is practically impossible to model a macroscopic system in terms of the microscopic kinematical and dynamical variables of all its particles. Thus one makes a hierarchical reduction in which this complexity is reduced to small number of collective variables. The theoretical framework for such reductions for systems is statistical mechanics or statistical physics.
One special case of hierarchical reduction is the limit of large volumes $V$, in which the number of particles (of each species) per volume, $N/V$, stays constant. This is called the thermodynamic limit in statistical physics. Under some standard assumptions like homogeneity (spacial and possibly directional) and stability (no transitory effects), there is a small number of collective variables characterizing the system. Such a description can be (and historically was) postulated as an independent self-consistent phenomenological theory even without going into the details of statistical mechanics; such a description is called equilibrium thermodynamics, which is believed to be deducible from statistical mechanics, as has been partially proved for some classes of systems. Sometimes transitional finite-time phenomena are described either statistically by studying stochastic processes or by a more elaborate hierarchical form of thermodynamics, so-called nonequilibrium thermodynamics.
One of the basic characteristics of a thermodynamical system is its temperature, which has no analogue in fundamental non-statistical physics. Other common thermodynamical variables include pressure, volume, entropy, enthalpy etc.
A formalization in terms of symplectic geometry is in chapter IV “Statistical mechanics” of
as well as in
Souriau model of Thermodynamics has been extented for Geometric Science of Information (Koszul information geometry) with a general definition of Fisher Metric, Euler-Poincaré Equation and variational definition of Souriau Thermodynamics, as given in:
See also
wikipedia: thermodynamics, fundamental thermodynamic relation
Azimuth Project, Thermodynamics
A. Bravetti, C. S. Lopez-Monsalvo, F. Nettel, Contact symmetries and Hamiltonian thermodynamics, arxiv/1409.7340
For an thorough introduction to common misconceptions at an elementary level:
A survey of irreversible thermodynamics is in
For more on this see also rational thermodynamics.
Some formal generalizations of thermodynamical formalism include mixing time and temperature in formalisms with complex time-temperature like Matsubara formalism in QFT.
Mathematical analogies and generalizations include also